Mathematical Sciences: Coherent and Chaotic Phenomena in PDE's

这些研究中涉及的数学涵盖了广泛的 可积和几乎可积理论的谱 系统和严格的分析偏微分方程的更直观的 在几何学的背景下最容易理解的思想。 在许多情况下， 特别是对流模式的研究， 光学，真实的实验作为指导，以开发和检查 模型的有效性。 在所有情况下，每个理论 调查伴随着广泛的数值实验。 一个互动的环境（这是显着增强， 许多科学游客）使亚利桑那州成为一个很好的地方， 学生的培训。 事实上，研究生们 参与本提案所述的研究。

The mathematics involved in these studies covers a broad spectrum from the theory of integrable and almost integrable systems and rigorous analysis in pde's to the more intuitive ideas best understood in a geometrical setting. In many cases, and specifically in the investigations in convective patterns and optics, real experiments are used as a guide to develop and check the validity of models. In all cases, each theoretical investigation is accompanied by extensive numerical experiments. An interactive environment (which is significantly enhanced by many scientific visitors) makes Arizona an excellent place for the training of students. Indeed, graduate students are heavily involved in the research described in this proposal.